Tuesday, November 24, 2015

Defending the New Natural Law's Incommensurability Thesis

New Natural Law (NNL) usually gets a bad rap from certain Thomists due to its embracing the "incommensurability thesis." I'll first explain the basic outline of NNL. I'll then explain the incommensurability thesis, and present an argument for it that I find quite convincing. Finally, I'll defend incommensurability from a common objection, and actually show how the objection favors incommensurability. Hopefully my defense will help illustrate the intuitiveness of the thesis.

First, a little explanation of NNL. NNL derives specific moral/practical norms by positing two things:
  • (1) A set of basic goods--things that are intrinsically good/valuable/worthwhile for humans to pursue. So, they are always "worthy of pursuit." In other words, they are always "to-be-pursued." Also, the basic goods can make immediately intelligible one's action, and are thus reasons for action. This is very important: The basic goods are the fundamental reasons for action; the fundamental "building blocks" of practical rationality. Examples include knowledge, life, friendship, aesthetic experience, and excellence in play.

    Let's consider an example. For instance, suppose you ask me, "Why are you running?" and I say, "To help my heart get stronger," to which you reply, "Why do you want to make your heart stronger?" If my answer is, "For the sake of my health, that I might live and live to the full," then this is a "question-stopper;" you can immediately see why I might want to do this, and thus health is a reason for acting that makes my action immediately intelligible; it immediately "make sense." This is because acting for the sake of someone's health is intrinsically valuable and worth doing.

    This is the first part of NNL: There is a set of basic goods, as defined above.
  • (2) A set of principles of practical reasonableness--these are principles or rules that tell us how it is reasonable to respond to instances of the basic goods. Examples include the golden rule, the prohibition against directly intending to harm a basic good for the sake of something else, rules against arbitrarily discriminating against the goods of certain people, the rule of being reasonably efficient in how one pursues one's plans, etc.
So, we start out with a list of things that are intrinsically good for us to pursue, and then there are rules that tell us how it is reasonable for us to go about pursuing them. The list of goods, in conjunction with the rules for practical reasonableness, ultimately tell us what actions are permissible as well as what actions are obligatory. They tell us what we may permissibly do, as well as what we ought to do. These rules are taken to be constitutive of practical reasonableness; they are supposed to follow from the very nature of practical rationality.

Much more needs to be said here obviously. Many books have been written. However, I don't need to get too much into the details of this for the sake of my post here. That's just the basic idea.

Now, one of the ideas about the basic goods that motivates some of the principles of practical reasonableness is the incommensurability thesis. Here is the thesis of incommensurability:
  • (INC) For each basic good x, it is not the case that x offers all the goodness ("to-be-pursuedness") of some other basic good and more.
See, for instance, Christopher Tollefsen's helpful paper. NNL theorists use the incommensurability thesis to present what I think is one of the strongest arguments against consequentialism of any sort.

Now, the motivation for the incommensurability thesis, as I've stated it, is that it seems that each good is beneficial to humans in a way which none of the other goods is. It has a unique type of goodness. Health is good for us in a way that knowledge is not, which is good for us in a way that friendship is not, and so on. If on the other hand we supposed the contrary of incommensurability, then some basic good x would offer all the goodness and more of some other basic good, say y. But then it's hard to see how y could make one's action immediately intelligible (which is part of the definition of a basic good). After all, if someone says, "I did this action for the sake of y," then in some circumstances you can reasonably say, "Well, you should have acted for the sake of x instead, since it offers all the goodness and more of y. So why didn't you?" In which case y would not be a basic good, since acting for the sake of y should make one's action immediately intelligible, and in this case it doesn't. But that's a contradiction, since we assumed y is a basic good. So the incommensurability thesis is true.

Now, one objection is that it seems we should be able to prioritize the goods. Religion is more important than life, life is more important than knowledge, knowledge is more important than fun, and so on. This seems plausible to some. But, if NNL is right, none of these can be "weighed" against the others, since none of them offers all the goodness of any of the others and more. So, the objection goes, if incommensurability holds, then we can't prioritize the goods.

Friday, November 6, 2015

External Reasons: A Natural Law Response to Williams

In my last post I criticized Bernard Williams' rather Humean argument in "Internal and External Reasons" for arguing that external reasons can't explain action. As a follow up, I wanted to post my paper I wrote for the meta-ethics class I'm taking where I criticize further Williams' argument against externalism, and at the same time build up a natural law account of reasons and practical rationality in opposition to Williams. Here it is!

                      In “Internal and External Reasons,” Bernard Williams presents an argument for thinking that external reasons do not exist, and thus all external reasons statements are false.[1] In this paper I will do three things. First, I will explain what Williams understands internal reasons and external reasons to be. I will then explain Williams’ argument against external reasons. Finally, I will attempt to give some defense of external reasons by critiquing Williams’ argument.

                      The general form of a reasons statement is “A has reason to do F in circumstances C.” Williams aims to show that statements of this form are only ever true on an internal interpretation. While Williams does not seem to give a definition of internal reasons statements, he does lay out what seems to be a necessary condition on internal reasons statements. He says that any internal interpretation of a reasons statement must “display a relativity of the reason statement to the agent’s subjective motivational set,” which we shall call “S.”[2] Roughly then, internal reasons for an agent are dependent on what is in the agent’s subjective motivational set.[3]

                      Williams also lays down as a necessary condition on internal reasons that they can be discovered by deliberative/practical reasoning.[4] While Williams does not explicitly define what deliberative or practical reasoning is, he specifies his conception of deliberative/practical reasoning via example. In particular, he says that practical reasoning includes means/end reasoning about the most preferable way of satisfying a desire, temporal ordering of when to satisfy which desires, determining which desires one is most interested in satisfying, and determining what would constitute satisfaction of one’s desires. So, this condition amounts to saying if A has an internal reason to F, A must be able to motivate herself to F by a process of reasoning of this sort (from S).[5]

                      It is clear then that Williams is working with an idea of internal reasons that ties them closely to an agent’s current subjective motivational set. It is also clear that he is working with a “thin” notion of practical rationality. Since Williams doesn’t explicitly define what he takes rational deliberation to be, it is difficult to precisely state what this “thin-ness” amounts to. However, roughly speaking, Williams’ account of rationality is “thin” insofar as, on his view, a decision will count as rational to the extent that it could be concluded to by a process of deliberation starting from one’s desires and satisfying certain (relatively weak) formal constraints.

Thursday, November 5, 2015

Bernard Williams on Internal Reasons (Contra Natural Law)

Here is a short precis I wrote for my meta-ethics class about Bernard Williams' argument for internalism about reasons. Internalism about reasons is, roughly, the Humean-ish idea that an agent has a reason to act a certain way only if it is somehow grounded in the desires of the agent. This is directly contrary to the natural law view, according to which what is reasonable to do or not to do is independent of the desires of the agent, and thus reasons are independent of an agent's desires:

In “Internal and External Reasons,” Bernard Williams presents an argument for thinking there is a problem with external reasons statements.[1]
1.    If something R can be a reason for action, then R can be a reason for someone A’s acting on a particular occasion O. [Premise]
2.    If R can be a reason for A’s acting on a particular occasion O, then R can figure in an explanation of A’s acting in O. [Premise]
3.    So, if R can be a reason for action, then R can figure in an explanation of A’s acting in O. [1,2 HS]
4.    If R can figure in an explanation of A’s acting in O, then R cannot be an external reason. [Premise]
5.    So, if R can be reason for action, then R cannot be an external reason. [3,4 HS]
6.    If R cannot be an external reason, then R is an internal reason. [Premise]
7.    So, if R can be a reason for action, then R is an internal reason. [5,6 HS][2]
Hence, the argument establishes that the only coherent notion of a reason for action is an internal reason for action, i.e., one which is relative to an agent’s subjective motivational set S. For now, let us grant premises 1 and 6. I’d like to think about premises 2 and 4 a bit. My point will be that, depending the sense of ‘explanation’ here, either one or the other premise will be implausible.
                      First, let us consider the interpretation of explanation where an explanation of an action is something that (directly) motivates someone to do that action. In this sense, maybe we should accept 4. For maybe it is only one’s desires, or at least something closely bound up with one’s desires, that could explain an action in the sense of being able to directly motivate it.[3]
                      However, in this case, I think that premise 2 is implausible; for I don’t think that a reason for acting in an occasion has to be able to motivate you. Rather, a reason for performing P in circumstance C is some state of affairs or consideration that could make performing P in C practically intelligible.[4] And something can do this without being able to motivate. Let me explain.
                      If you found me in a chair stabbing myself in the leg you would ask me why I’m doing what I’m doing. If I said to you, “I just desire to,” this wouldn’t really answer your question. There is still a clear sense in which what I’m doing is unintelligible, and wouldn’t make sense even if I obeyed some bizarre psychological laws which led me to “just desire” to stab myself in the leg. If however I said, “I am scared there is a tiny alien in my leg, and I need to get it out so it doesn’t kill me,” I will at least have described some state of affairs where, when my action is seen in light of this, my action does at least become intelligible (even if, in all likelihood, it is still utterly ridiculous and unreasonable). Such a state of affairs is a reason for acting a certain way.
                      But if this is a correct account of what reasons for action are, then 2 seems false. For it might be that a state of affairs S could make doing P intelligible, and thus counts as a reason for doing P, but my subjective motivational set makes me completely unable to even take S into consideration, and thus makes S unable to motivate me. This happens all the time. For instance, suppose I have no desire or motivation to quit smoking. However, the fact that smoking greatly increases my chances of dying certainly could make the action of quitting intelligible (i.e., if someone appealed to this fact when justifying their choosing to quit). So the fact that smoking greatly increases my chances of dying is a reason to quit. Yet, again, it might be that it couldn’t motivate me on any occasion.
                      All of this is to say that, on the interpretation of ‘explain’ where it is roughly equivalent to ‘motivate’, premise 2 seems implausible. On the other hand, if ‘explain’ means ‘make intelligible’, then premise 2 seems true, but premise 4 seems false, at least if reasons for action are, as I’ve explained, states of affairs which make acting certain ways practically intelligible.

[1] The “problem” is either that they are false, incoherent, or are misleading (and presumably should instead be rephrased as internal reasons statements). See the last paragraph of p. 297.
[2] This is my interpretation of Williams’ argument at the bottom left paragraph of p. 295. I may be misinterpreting Williams, but this seemed to me the best way to make his argument valid.
[3] I am assuming the sense of ‘can’ here means something along the lines of ‘is physically possible’.
[4] This account is largely adapted from G.E.M. Anscombe’s book ‘Intention’, sec. 37.

Friday, October 30, 2015

Ineffability: A Serious Threat to Ambitious Metaphysics

Consider a squirrel on a tree. There are things we can represent that are simply not within the capacity of the squirrel's mental structure to represent. For instance, we can represent complex mathematical truths about the shapes and physical relations obtaining in the squirrel's environment. Despite the fact that all these facts are "happening" right in his face, our little squirrel hasn't the slightest clue about them. And it's not just that he's ignorant about them like someone who doesn't know math or physics; it's that it's completely beyond a mind like his to even represent things like facts and propositions. They are completely ineffable to the squirrel.

That all seems fair enough. But here's where things start to become worrisome: Why shouldn't we think that we are in relation to some possible being in the same way that the squirrel is related to us? (Or, if you believe in God, why not think that we are related to some actual being in the same way that the squirrel is related to us?) In other words, why not think that there are some aspects of reality that are completely ineffable to us, even if maybe they are effable with respect to some other, more advanced being? Why not think that there could be some species of creature who could comprehend things that are completely beyond our mind's capacity to even represent?

This is not a new argument or anything. See for instance Ch. VI of Thomas Nagel's book 'The View From Nowhere'. Moreover, I'm sure many of us have thought of this possibility before. But do we really consider the implications of this argument? I know that I've thought of this before, but haven't drawn anything of significance from this. However, I'm realizing now that this is a very important and deep question.

Note that this view seems to go hand in hand with the idea of metaphysical realism, which holds that there is an objectively existing world that is in some sense totally (or at least mostly) mind-independent. If you hold that view, it seems rather strange to think that somehow all of reality, of necessity, must be in principle comprehensible to us. And if reality were in its totality to be comprehensible to us it would be a rather strange coincidence. Moreover, there seems to be nothing particularly special about us. We are on a continumm with non-sentient creatures, insects, and squirrels on one side and with angels and God on the other (and probably a lot of things in between). Hence, it seems quite likely, on realism, that we are in a situation similar to that of the squirrel: There are aspects of reality which are simply beyond the representational capacities of our minds.

But then there is trouble. Interestingly, those who have high hopes for metaphysics tend to be metaphysical realists, but that very same metaphysical realism tends to undermine the high hopes for metaphysics. For example, suppose we characterize metaphysics as the study of the most fundamental or general aspects of reality. Suppose moreover that we are metaphysical realists. Then, probably, there are aspects of reality which are simply beyond the representational capacities of our minds. But in that case, for all we know, the most fundamental or general aspects of reality are within the sphere of things that are completely ineffable to us. So there is reason to doubt the possibility of having any substantial metaphysical knowledge.

In fact, maybe by a similar though distinct argument we can get a stronger conclusion. We have some reason to think that as minds become more advanced on the "great chain of being" that I've described, they become able to represent more (and more) fundamental aspects of reality than those before them. Higher beings have concepts that are more fundamental than those of the minds on lower levels. (Technically, they have concepts of things that are more fundamental.) And they probably have more of them. For instance, some lower animals can probably represent things like 'cause' and 'object' and even 'agent', but it seems doubtful whether bees could do the same (or to the same degree). But in that case, granted we are probably pretty far from the high end of the continuum, probably the most fundamental aspects of reality are only representable by beings on the higher end. So, probably, we cannot represent the most fundamental aspects of reality. So, probably, ambitious metaphysics is hopeless.

This might seem like a fun philosophical puzzle, but actually it is rather important, because if I sit down and ask myself whether I really think metaphysical realism is true, I am with utter and literal sincerity inclined to say, "Yes." And if I sit down and ask myself whether I really think some aspects of reality are ineffable for the reasons described, I am with utter and literal sincerity inclined to say, "Yes." And, to bring the trilemma to completion, I have high hopes for metaphysics and sincerely think it is essential to truly understanding the world.

What to do then? Does the argument against substantial metaphysics work? What are the implications for metaphysics and other areas of philosophy depending on which way one goes? How might different views solve the issues here?  These are interesting questions. Since I've been thinking about this stuff for a class I'm taking, I'll probably have a chance to write a term paper on it. I have inklings about where we might go, but I have no clear answer at the moment.

Friday, October 16, 2015

Link: Amazing Philosophy Interviews with Bryan Magee

I just wanted to share a very special link with any readers who happen upon my blog.

Several years ago (probably about five or so years ago) I worked as a mascot for Liberty Tax Service in southern California. At the same time I was going to community college, and I had decided by this point that I wanted to study philosophy. I had only begun to learn about analytic philosophy. While waving a sign dressed up in a statue of liberty costume can be fun at first, after several hours one might feel a need for intellectual stimulation. Hence, I would download philosophy talks from several different sources, and one of them was here.

There are many full-length interviews on this Youtube channel, all available for free. They each consist of a dialogue between Bryan Magee and one of the greatest philosophers of the twentieth century. They are very interesting, and are especially helpful in the case that one is not already familiar with the concepts being discussed (though even then they are still fun to listen to). If you're not familiar with some area of philosophy they can give you a quick crash-course on the topic. I really recommend checking them out. Here is one I remember enjoying a lot, with A.J. Ayer talking about logical positivism. One of the best lines is when Magee asks, "What do you think was the biggest issue with logical positivism?" and Ayer replies, "Well, it was all false." (Part 4, 6:26)

Friday, October 9, 2015

Greg Cavin on Bayes' Theorem and Miracles

I wrote most of this post several months ago when my friend Calum Miller came to southern California for a semester abroad. Unfortunately, I simply never got around to finishing it up. Hence, this post comes about five or six months late. However, I still think it's worth posting, in case someone watches the video or comes upon the type of fallacy that I suspect goes into the argument. Here's the post:

A couple weeks ago I went to a debate between my friend Calum Miller and philosopher Greg Cavin on the Resurrection of Jesus. The video can be found here. Cavin's opening speech on Bayes starts at (6:00). He gets into his arguments again at (13:40). In this post I'll discuss a small part of Cavin's opening speech.

At the beginning, Cavin claims that he will show that it is "virtually 100% certain that no miracles ever occur."

Greg Cavin formulates the argument in terms of an assessment of a comparison between probabilities. While Cavin goes into a ton of mathematical detail that I suspect could be simplified to get to the main point, a little bit of it is probably necessary. He formulates the argument in terms of the Odds Form of Bayes' Theorem.

In general, the Odds Form of Bayes' Theorem is as follows. For any events A, B, and D:

P(A|D)/P(B|D) = P(D|A)/P(D|B) * P(A)/P(B)

Cavin comes up with a partition of probability space which is exhaustive and exclusive. In other words, at least one of the following hypotheses holds and if one holds then the others do not.

  • M: At least one miracle has, had, or will occur in the universe.
  • L: The laws of the sciences as these are currently formulated in standard reference works, without any supernatural non-interference proviso, are true and are laws of nature in their restricted domains.
  • (¬M & ¬L): Neither M nor L hold. 
With respect to L, what it is saying is that if you have a law of science 'S,' then a statement of the law will just be of the form: "For all times, all places, S," rather than "Except for the intervention of some supernatural force, for all times, all places, S." In other words, laws of science lack the underlined "proviso."

M and L are taken to be incompatible because if a miracle occurs, i.e. if M is true, then that entails the failure of at least one "un-provisoed" law of science at some time and place, whereas L entails all "un-provisoed" laws of science hold at all times and places. Cavin defines the evidence E with respect to which we will evaluate these probabilites as follows (27:00):

  • E: The total evidence, which is a combination of T & C, where T and C are understood as follows:
  • T: All of the traces (call them Ti) of miracles. These are all of the pieces of evidence people could take to provide evidence for a miracle.
  • C: All of the confirmation instances (call them Ci) of the laws of science. These are all of the pieces of evidence people could take to provide evidence for the various scientific laws.
The partitioning of probability space.

Applying Bayes' Theorem to the argument at hand, this is the Ratio of Posterior Probabilities of L vs M:

P(L|E)/P(M|E) = P(E|L)/P(E|M) * P(L)/P(M)

In other words, the left hand side compares the likelihood of L given the evidence with the likelihood of M given the evidence.

Now, a crucial part of Cavin's argument is in calculating the ratio P(E|L)/P(E|M). This is done by calculating probabilities of all of the Tgiven L and M and calculating the probabilities of all the Ci given L and M. If these are lower on M than they are on L, then P(E|L) will be higher than P(E|M). His official argument here is from (31:00) to (34:00), but I asked him a question later that gets to the same point.

After the talk, I asked Cavin why he thought M could not explain C and T as well as L could. In other words, why are, say, the confirmation instances of science less likely given that miracles have occurred than if L holds? He said, "Well, if I told you, 'This is a desk,'  what would that explain? Not much. How can you make any predictions from that? So, likewise, how can the proposition that at least one miracle holds explain anything? It could hardly have any predictive or explanatory power." Of course, that seems true. If the only sentence you knew to be true were "At least one miracle occurs," then you wouldn't be able to predict much, just as you couldn't predict much from just knowing "This is a desk." Hence, the argument goes, P(E|M) is very low.

However, it's a little bit misleading to put things this way. P(E|M), strictly speaking, isn't defined in terms of how much you can predict from the single proposition that at least one miracle occurs. This is clear after considering some very basic probability theory.

First, note that we can always define P(M) as P(MA) + P(M∩¬A) for any event A. This can clearly be seen by the following diagram:

P(M) = P(MA) + P(M∩¬A)
A is marked out in dark blue.
¬A is marked out in light blue.

Suppose that 'A' denotes some hypothesis, maybe the hypothesis 'The laws of nature almost always but not always hold.' Then the probability that some miracle happens is equal to the probability that some miracle happens and A holds plus the probability that some miracle happens and A does not hold. Again, P(M) = P(MA) + P(M∩¬A).

From this we can infer: P(E|M) = P(E|MA) + P(E|M∩¬A). Now, you might still think that this is lower than P(E|L) for various reasons. But you certainly couldn't infer it from the type of argument I sketched above. That would be much too easy.

Maybe I am misrepresenting what Cavin said. I hope I'm not. But if I am, let's just say that if someone were to argue in the way I represented Cavin as arguing, then they would be committing a fallacy.

There were many other interesting issues that came up during the debate, such as the likelihood of the laws of nature holding most of the time given theism, and these deserve attention. But for now I think it's worth noting that Cavin's argument doesn't go through as easily as it might have seemed.

Tuesday, October 6, 2015

Shameless Hyperintensionalism in Ethics (And Other Areas)

A hyperintensional position in a sentence is one where substitution of necessarily co-extensional statements does not preserve truth value. So for instance, 'believes' is a hyperintensional position. Alfredo believes triangles all have three sides doesn't necessarily imply Alfredo believes that triangles all have angles adding up to 180 degrees (and vice versa). After all, I might not know this yet.

What I will call a metaphysical hyperintensional position (and which I will just call 'hyperintensional' hereafter) is, intuitively, one where the hyperintensionality doesn't arise because of some mental attitude. This can be defined more precisely, but for my purposes some examples will suffice.

For instance, the operator 'essentially' is hyperintensional, at least on one understanding of 'essentially.' To use a historical example, Socrates is essentially a rational animal; he is not essentially risible, though necessarily if something has the one property it has the other. Or to use the more contemporary example, Socrates is not essentially a member of his singleton set -- this doesn't have to do with what he is, at the most fundamental level, in himself -- even though he has the property of being so necessarily. Grounding, intrinsicality, naturalness, reduction; these all seem to be hyperintensional as well.

Many metaphysicians are skeptical of these concepts. (Though of course many are not! Which is why they are such a hot topic of discussion lately.) I think there is usually some ambiguity in what this "skepticism" amounts to, but it is often made explicit in terms of the good old, "I don't know what that means." This skepticism looms especially large among those of a certain breed of metaphysician, whose generation either made advances over extensional concepts by employing intensional ones, or learned from those who did. (On all of this, see Daniel Nolan's very enjoyable paper.)

One thing I find worth noting is that people in ethics use these concepts shamelessly, both in first-order normative ethics and in meta-ethics. Meta-ethical claims are consistently stated in terms of "in virtue of" (just witness the Euthyphro Dilemma) or "grounding." There is a consistent flow of talk about what is essential to an action (as opposed to what is just necessarily true of it), as well as the intrinsic features of an action. Ethicists seek real definitions of what's right and wrong, seek to categorize things into natural kinds, seek to reduce properties to other ones, and seek to explain less fundamental facts in terms of more fundamental principles; in general, ethicists have no qualms about using metaphysical hyperintensional concepts, while many metaphysicians claim skepticism even about their intelligibility.

Philosophers of mind initially seem to be a bit more careful in this regard, at least those working in the metaphysics of mind. Witness the debates about supervenience for instance. However, it seems many philosophers of mind are realizing that they have really been trying to raise issues that can only be adequately stated using hyperintensional language. And many times even those who are keen to phrase things in terms of supervenience will explicitly distinguish this purely modal notion from what's really doing the work (explanation, grounding, reduction, and so forth).

Many perfectly legitimate metaphysical debates themselves are best phrased in terms of hyperintensional concepts. And if we look in areas other than metaphysics, a very similar pattern seems to arise; think of philosophy of science, philosophy of math, philosophy of religion, free will and moral responsibility, etc.

One of the clearest examples though is in ethics. It might be worth it for me to demonstrate my empirical claims with specific examples (though frankly it's harder for me to think of papers in ethics which don't make any use of these concepts than those which do). I might try to do this at some point. But I think it's just worth pointing out for now: The sense I get from everything I've read in ethics over the years is that ethicists have no problem whatsoever using any of the hyperintensional notions that metaphysicians will sometimes claim to have no understanding of. Obviously that is not universally true, and some positions in ethics are in fact more "deflationary" than others. But arguably this is not the norm.

Considering that ethics is often one of the closest areas of philosophy to "real world" problems this is important, since it lends some weight toward thinking that these are not arcane, idiosyncratic notions, but would be considered completely intelligible to most people (and they are; anyone taking intro to ethics will "get" the Euthyphro Dilemma; in fact, their understanding of this problem will probably be much clearer than any problem involving modality for example). Also, if you grant a certain level of epistemic autonomy to ethics -- it doesn't need to wait on the approval of metaphysicians to be considered legitimate -- then it would be wrong to pronounce, based on an ill-defined "skepticism," that these concepts are unintelligible. The interaction between metaphysics and ethics should really be one of reflective equilibrium. And in that case, the initial reaction to hyperintensional concepts should be one of cautious approval rather than default skepticism.

Friday, October 2, 2015

Review: 'An Aristotelian Realist Philosophy of Mathematics' by James Franklin

I recently finished reading James Franklin's marvelous book, An Aristotelian Realist Philosophy of Mathematics, and I want to advertise it here. This is a great book. It is empirically informed by a wide knowledge of both actual mathematical practice and contemporary mathematics itself, along with other relevant areas of study such as perceptual psychology, neuroscience, and engineering. It also engages with much of the cutting edge in contemporary philosophy of mathematics, especially in the later chapters. This is some of the best of what Aristotelianism has to offer. I really hope people will read it.

Franklin aims to give an account of mathematics as the science of quantity and of structure. Franklin gives particularly clear definitions of both quantity and structure--something often lacking among contemporary structuralists in my opinion--and this in itself is a very important advance. According to his account, mathematics studies structural universals and quantities. These universals and quantities are the type of thing that can be found in the real world and can be literally had by concrete objects. Of course, not all mathematical structures are had by some concrete object, but it is essential to his account that they could be, i.e. that they are metaphysically possible.

While quantity seems to me to play a less central part in his project, his clear account of structure allows him to take his views a long way. Franklin understands a property to be purely structural just in case it can be defined completely in terms of 'part', 'whole', 'same', 'different', and purely logical vocabulary. The relations of 'part' and 'whole' will probably come into play in geometry, as well as set theory, graph theory, topology, analysis, etc. So, for instance, on this definition, the property of being a Euclidean space could probably be defined purely structurally; see for instance Hilbert's axioms. Also, the Peano axioms seem to describe purely structural relations, since they only invoke logical vocabulary and identity (other than the names for the relations being defined, of course). Franklin gives many more examples, so I refer the reader to his book for a treatment of further cases.

Franklin contrasts his approach with Platonism and nominalism in contemporary philosophy of mathematics. Unlike Platonism, the universals studied by mathematics can be literally instantiated by concrete things in the real world. What mathematics does is study these possibly instantiated structures. Mathematics does not study abstract, particular individuals. Number systems, for instance, would not be cashed out as consisting of abstract individuals (numbers), but as either systems of quantities or as structures which can be instantiated by concrete things. (Franklin's account of number, in fact, cashes out numbers as being relations which are literally instantiated in the world by material heaps and 'unit-making' universals.)

Against nominalism on the other hand, Franklin assumes that there are, in fact, mathematical universals that can be literally shared by different things. Again, Franklin also assumes that there are, in addition to those universals instantiated in the real world, universals which are not instantiated but are at least possibly instantiated.

By his choice of example he shows how contemporary philosophers of mathematics often miss the most central cases of mathematics. Contemporary philosophy of mathematics often has a Platonist bias, focusing on those cases that are less essential for use in real world applications (such as huge sets, large infinities, etc.). This is to the detriment of the most central and basic cases, which are the simple, often discrete and finite structures widely used in real-world applied sciences, and which are less amenable to Platonist interpretation.

He gives a far more plausible account of mathematical knowledge and empirical mathematical application than that offered by most Platonists. He also argues that contemporary philosophy of mathematics tends to not pay enough to attention to how mathematics is actually done, and therefore misses those aspects of mathematical practice that make more sense on an Aristotelian view. He shows a much closer parallel between actual mathematical practice and actual empirical scientific practice than is often recognized (for instance, by the unquestionable use of induction, plausible reasoning, and explanation in mathematics; he rightly notes that (in)formal proof is often only the last step in the equation). Franklin goes on to apply the Aristotelian conception of mathematics to many other philosophical issues, such as mathematical necessity, infinity, approximation, and ontology.

With that said, there are several parts of the theory that could be potentially problematic and call for more investigation. Just to shotgun a few of them out:
  • The reliance on a classical mereology of heaps and arbitrary sums (this is important for his definitions of whole numbers and sets).
  • The reliance on (immanent) universals, problematic from a trope nominalist perspective such as my own, and which might use a bit more explanation.
  • The commitment to uninstantiated universals (an idea classically denied by most Aristotelians, including Aristotle himself, and one which moves Franklin's account toward a "semi-Platonism" as he calls it).
  • His commitment to all mathematical structures being metaphysically possible (this is interesting to me; I bet Franklin's account could be seamlessly extended given a proper account of impossibility, impossible objects, impossible universals, and impossible worlds, and I bet this isn't essential to his view).
  • Giving a general, unified semantics for mathematical language (it's less than clear from the book how this is to be done; for instance, with the complex and negative numbers, Franklin gives what appear to be examples, or maybe geometrical/economical interpretations. But what would he say are the straight up truth-conditions for, say, -2 + 3i = 2(-1 + 3/2i)? Or (-2)(-3) = 6? Or of more general laws governing number systems?).
  • Showing more precisely and in individual cases how a more wide range of mathematical concepts are definable either purely structurally or quantitatively (ideally, it'd be nice if we could get to the point of giving a general paraphrase scheme or a general procedure--Franklin's account of set theory being purely structural is suggestive, so maybe we could show how any set-theoretical entity or relation could be defined structurally, and thereby show all mathematics to be interpretable structurally; either this or the last question I hope to work on for my term paper this semester).
  • The apparently ad hoc fictionalist account of zero and the empty set combined with a realist account of everything else (I can see fictionalists asking why we need the realistic ontology in some cases but not others).
  • Related to this last point, some unclarity/implausibility in the theory of ontology and ontological commitment at play, as well as some unclarity about the ontological status of mathematics (if it were made more clear when or why we are committed to some things but not others, and in what way, it'd probably be easier to answer questions such as the last one).
I don't have enough time to spell all these worries out, though if anyone is curious I can explain what I mean, and maybe after reading the book some of these worries will be clear. And I don't think these are damning or insuperable criticisms either; I think they are problems to be investigated, but Franklin's account seems to me to be certainly on the right track.

One last potential criticism that I feel kind of bad about making: I feel like the book doesn't really engage much with what's been said in contemporary neo-Aristotelian metaphysics and ontology. I feel bad about saying that because of the huge swaths of literature the book does, in fact, engage with (the number of works referenced is amazing; one wonders how somebody can read so much). But in certain respects (the mereology for instance, or the role of states of affairs), it seems like the book draws on some concepts with which many current Aristotelians might take issue. And like I said, the book's understanding of ontological commitment could have been a bit more clear; here, engagement with contemporary Aristotelian metaphysics (among others) might have been helpful as well.

Overall though this is an excellent book, and maybe even a game-changer, at least for me. It contains many more interesting ideas and arguments to grapple with than I've been able to discuss here. Whether one buys into it or not, Franklin admirably demonstrates the fruits of an Aristotelian approach, at least on one understanding of that term. He makes use of a wide variety of examples, from a wide variety of real world sciences (including, but very much not limited to, pure mathematics). By doing this he demonstrates how important it is to pay attention to actual empirical results and practice when doing any sort of metaphysical or epistemological investigation into the philosophical status of mathematics. And this seems to me to be one of the most important marks of the general Aristotelian attitude.

Wednesday, September 30, 2015

Link: Pope Francis: Marriage is Indissoluble

Pope Francis states, in very clear terms, the traditional Catholic teaching on the indissolubility of marriage, here.

Quote: Foot Making Fun of Expressivists

In her book 'Natural Goodness' Philippa Foot criticizes (lightly mocks) expressivist accounts of moral evaluation because they seem to make evaluation of human action completely disconnected from our evaluation of other biological aspects of human well-being, as well as the evaluation of the goodness of other kinds of animals and plants.

"For it is obvious that no expressivist account will do in those other domains: we cannot think that the use of the word 'good' is to express a 'pro-attitude' in what we say about the roots of nettles or the fangs of ferocious beasts. Nowadays such evaluations are apt to be marginalized as if they were fanciful extensions of the 'proper' evaluations that express our attitudes, practical decisions, or desires. But when I was told by a certain philosopher who wanted to explain 'good' in terms of choices, that the good roots of trees were roots of the kind 'we should choose if we were trees', this finally confirmed my suspicion of the kind of moral philosophy that was his."

Tuesday, September 29, 2015

Libraries: A Case of Practical Incommensurability

In debates in ethics (among new and old natural law theorists for instance) one problem that comes up is the incommensurability of certain types of goods. What this means is that it doesn't seem that in general we can even weigh certain goods against others (for example, say, aesthetic experience and friendship). An apparent problem for utilitarians and consequentialists of various stripes, among others.

I'm not sure how much this has to do with that, but it struck me now as I'm working in my personal library of books how much incommensurability considerations affect me on a concrete, everyday level.

I was sitting here looking over a syllabus, and I realized I will have a week-long break or so in a couple weeks, and I'm going to probably pick a book or two to read over that break. Thinking about that, I realized that at a practical level, I have so many books to choose from that I probably will not read all of them straight through any time soon (if I'm honest with myself, maybe even ever). So it'll be a tough decision. I'm probably going to spend at least half an hour going through my long series of unread books and deciding which category of philosophy to even read from (if I can get myself to the point of deciding to read philosophy instead of something else). Then, supposing I've chosen a category (say, ethics), I'm going to have to decide which of several monographs within that category to choose.

One might think that when I'm choosing which book of ethics to read there must be something which marks one of the books off as more worth reading than the others. But I don't think so; I'm probably just going to have to pick one among equally reasonable options. Assuming the authors I'm considering are all equal on obviously measurable standards such as clarity in writing, intelligence, standards of rigor, etc., what other criteria of goodness would there be here? Quality of subject matter? How can I weigh that? If not that, what else? And even if I were to grant that I do a straightforward weighing along a single or several variables, this suggestion seems much less plausible when I'm making the higher-order choice of which philosophical area to read in (or the choice whether to read philosophy at all!).

You might then think that instead of saying my choice was among incommensurables I had multiple choices with an equal degree of goodness along all the variables of goodness, and my choice was arbitrary. In other words, the choices are commensurable, but they just happen to all be equal on the scale of goodness. But as I mentioned above, the subject matter of the books for instance does seem to play into my decision (at least in my case; I don't know about everyone else), and yet this doesn't seem like something I can literally weigh against the alternatives. Probably other incommensurable considerations play into my decision too.

(Note: Making a choice based on considerations and choosing one among alternatives doesn't by itself imply that the alternatives are commensurable. In fact, even if one choice is more rational than another, that doesn't imply the goods chosen are commensurable with those forgone qua goods. There might be some other considerations, for instance purely about what's constitutive of rationality, that bears on what is to be done.)

Maybe this is a case of incommensurability even within a particular category of basic good (viz. knowledge). I wonder how a utilitarian picture of everyday deliberation of this sort would go. And thinking about my own experience, I wonder whether a utilitarian could give a plausible model of such experience.

Okay, that's enough of that. Back to work.

Thursday, May 21, 2015

Essence and Counterpossibles

In my last post I made the point that the predicate position in real definitions is hyperintensional. So, even if two predicates have the same intension, i.e. necessarily apply to all the same things, they might not be able to be substituted for each other in the real definition while preserving truth value. This means that, among the necessary properties of a thing, we have to distinguish those which are essential to the thing from those which are not.

Maybe one way to do this is by using counterfactuals with impossible antecedents, also known as counterpossibles. 

The general idea is this. Counterpossibles, according to a certain semantics, are also hyperintensional. You can insert intensionally equivalent antecedents into the same counterfactual, but only some of these counterfactuals will be true while others will be false. So maybe we can use a particular counterpossible schema (as we will see, that in (1*)) to discriminate between properties that are essential and those that are non-essential. 

More specifically, we can take two intensionally equivalent properties F and G, insert F into the antecedent, insert G into the antecedent, and the counterfactual may have a different truth value depending on which of F or G is substituted. Thus, the essential vs. non-essential distinction will be able to be defined in terms of the truth or falsity of instances of a certain counterfactual schema. To make the point more vivid, the counterfactual schema will be like a box we can insert intensionally equivalent properties such as F or G into. If we put in G for instance and the box outputs TRUE then G is essential; if it outputs FALSE then G is not essential. Since counterpossibles are hyperintensional, the 'box' won't always give the same output for properties with the same intension.

So: Consider two intensionally equivalent properties F and G. This means that, necessarily, if anything has F in a world then it also has G in the same world, and vice versa. If F and G are intensionally equivalent then the properties λz[□Fz] and λz[□Gz] are also intensionally equivalent, and thus the formulas λz[□Fz](x) and λz[□Gz](x) are intensionally equivalent. Now, consider a counterfactual of the form:

  • (C) If φ had been the case then A.
Since we are going with an interpretation allowing for non-trivially true counterpossibles, we can substitute in for φ either of two necessarily false propositions, P or Q, but it won't automatically follow that (C) will come out true under both substitutions.

So suppose it is true that a is necessarily F and necessarily G.

Let P be '¬λz[□Fz](a)', let Q be '¬λz[□Gz](a)' and let A be '¬λz[∃yy=z](a)'.

Again, it doesn't follow automatically from the semantics of counterpossibles that substituting P in for φ will give you the same truth value as substituting Q for φ, despite the fact that these two formulas are intensionally equivalent. What we can do then to distinguish whether F is essential or G is essential (or neither) is substitute in 
P for φ and Q for φ in (C). If (C) comes out true, then the property is essential; if it comes out false, then it is not.

With this hypothesis in mind, here is a very rough first stab. For any x:

  • (1) For any P, if P is a de re necessary property of x, then P is an essential property of x if and only if had x lacked P then x would not exist.
More formally, let |x| be λz[z=x]: 
  • (1*) For any P: If λz[□Pz](x) then: |x|Px iff (¬λz[Pz](x) □→¬λz[∃yy=z](x)])
Again, keep in mind that the counterfactuals here are to have a semantics where they can be read as non-trivially true counterpossibles, and thus not according to the standard Lewis-Stalnaker semantics. Also, the 'essentialist box', F, comes from Kit Fine's 'Logic of Essence'. Roughly, FA means that in virtue of the essence of the F's, A holds. So if |x| is the property of being identical to x, then |x|Px turns out to mean that in virtue of the essence of x P holds of x. Fine parses out the logic this way for its semantical and logical elegance.

Consider one example given by Kit Fine: Suppose we have the property λz[z∈{z}]. Intuitively, this is the property satisfied by something whenever it is in its singleton. Assuming that, necessarily, I am a member of my singleton, then this is a de re necessary property of me, i.e. λz[z∈{z}](a) However, it seems to not be an essential property of me.

So, is it true that the following holds?

  • (A) Had Alfredo lacked λz[z∈{z}], Alfredo would not exist.
  • (A*) ¬λz[z∈{z}](a) □→ ¬λz[∃yy=z](a)
It seems that (A) does not hold. For it's irrelevant to my nature whether there are any abstract objects, such as sets, at all. After all, it seems that if nominalism were true, I would still exist. This lends some weight toward thinking that, if I had lacked λz[z∈{z}]I would still exist. But then by the criterion in (1), since the counterpossible does not hold for the property λz[z∈{z}], it must follow that λz[z∈{z}] is not essential to me. 

Given a de re necessary property of x, P, it might also be that the truth of the appropriate counterpossible is just a necessary condition for a property's being essential. In other words:

  • (NEC) If P is essential to x, then were x to lack P x would not exist. 
  • (NEC*) If |x|Px, then (¬λz[Pz](x) □→¬λz[∃yy=z](x)]).
Or it might be a sufficient condition as well, i.e.:
  • (SUFF) Given that if x were to lack P x would not exist, then P is essential to x.
  • (SUFF*) If (¬λz[Pz](x) □→¬λz[∃yy=z](x)]), then |x|Px.
Keeping in mind of course the sense of the term 'essence' in mind, and the relevant semantics for counterpossibles, (NEC) seems definitely true, and probably uncontentious. I suppose the interesting question is whether (SUFF) is true. I think the examples lend some support to the idea, such as the case of the singleton set given above.

In the case of (SUFF) it is particularly important that we use the right semantics for counterpossibles. If (SUFF) is true then this is very useful when talking to those who don't recognize the sense of essence at stake here: If they already know how to evaluate the counterpossible in the antecedent according to a 'non-trivial' semantics, then we simply say, "Plug in the property for the antecedent; if the counterfactual holds non-trivially, the property is essential. Now you know what I mean." This might also be a nice way to interpret people who give multiple definitions of 'essential' and 'accidental' properties, such as Aristotle. Aristotle gives a modal definition of essential properties which can sound like it might contradict other definitions of his; but (1) is 'modal' too, and it might be a way to interpret Aristotle that makes him consistent.

Probably potential counter-examples to the hypothesis come to mind. I know that I already see some issues. But it might be useful to see how far this hypothesis can go. And maybe if the hypothesis doesn't hold in general (I bet it doesn't) it might at least help us pick out an important class of the essential properties. After all, it seems in part that the reason we recognize λz[z∈{z}] as non-essential to me is because in some (non-trivial) sense had I lacked it λz[z∈{z}] I would still exist. Had nominalism been true, I'd have still been real (save if you're a Platonist/Pythagorean of a certain sort, in which case maybe you'd have good, non-trivial reason to deny that the counterfactual holds).

It might seem overly complicated to do this quasi-formally as I have, but one thing I'd like to do is look more at the formal semantics of essence such as Fine's for instance (hence the essentialist 'box' operator from Fine's papers), and see how this relates to the formal semantics of counterpossibles (whatever that happens to be). 
Given the inter-dependence of the two notions, maybe the correct semantics for counterpossibles will help us find the correct semantics for essence, and vice versa. Maybe the notion of essence will help us give a more principled similarity metric for counterpossibles in certain contexts. Also, maybe the notions of essence and counterpossible will relate closely to other notions, such as grounding, dependence and explanation. It might be an interesting project to see how far formal methods can help us here in finding relations between these concepts.

Thursday, April 16, 2015

Essence and Hyperintensionality

The essence of something is the truthmaker of the real definition of the thing. So, to know what the essence of something is is to know its real definition. For instance, to know the essence of man is to know the proposition that man is a rational animal. This is traditionally thought to be the real definition of 'man'.

Here is the general schema for a real definition:
  • S ise an F.
'S' is replaced by some kind-term (or maybe even individual-term?), the thing to be defined, and 'F' with some predicate, the definiens. The 'is' here is a special kind of 'is': the 'is' of real definition or essence. The conditions that have to be met for something to bee F are much more strict than for something to be F in other senses of 'be' (such as the more general sense of 'is', the 'is' of predication).

(Side-note: In some contexts is this a schema for reduction too? Interesting...)

Real definitions are 'fine-grained'. You cannot always substitute extensional equivalents into the predicate position to get the same truth value. For instance, suppose all and only the actually existing rational animals are animals which evolved by a certain evolutionary process P on earth. Even if this so, the following is not true:
  • Man ise an animal which evolved by process P on earth.
After all, man could have evolved in some other way, or even not at all. Man could have randomly popped into existence. So it's certainly not part of the very definition of man that he evolved by a certain evolutionary process.

So real definitions are fine-grained. In fact, real definitions are very fine-grained; you cannot even substitute intensional equivalents into the predicate position and always retain the same truth value. Suppose for instance that, necessarily, any animal which is rational is the type of thing which can speak a language. This actually seems pretty plausible. (If not, think of some other necessary consequence of being rational. You could even use some fancy disjunctive, conjunctive, or conditional properties, though I try to avoid these.) Even if this is so, the following is not true:
  • Man ise an language-capable animal.
At least, it's not true when we're talking about the 'is' of real definition. For this doesn't get to the heart of what man is; it's not what he is at the most fundamental level, but rather something he happens to be.

So, the predicate position in real definitions is a hyperintensional position, in the sense that substitution of intensional equivalents will not always preserve the same truth value. I take it these points cohere well with what has been said about real definition and essence up to now by others, such as Fine.

In the next post, I'll try to say something about how the hyperintensionality in real definitions means that counterpossibles will be very closely related to real definitions. Maybe this will help, at least a little, with the epistemology of essence.

Lately I have been suspecting that hyperintensionality, counterpossibles, essence, explanation, grounding, reduction, fundamentality, naturalness, intrinsicality, and lots of other things are very closely related. In the future I'd like to try to bring out some of these relationships. I'm not sure how successful this will be, but my metaphysical nose is leading me in this direction.

Friday, April 10, 2015

Modal Realism and the Serviceability Argument

Here's a quote from David Lewis: "Why believe in a plurality of worlds? -- Because the hypothesis is serviceable, and that is a reason to think that it is true."

Question for David Lewis and other modal realists: Lots of worlds are serviceable, not just the metaphysically possible ones. Many times when we do semantics, discuss language, give thought experiments, etc., worlds which are strictly logically possible but not metaphysically possible are helpful. For example, one of the ways that intensional semantics deals with oblique transitive verbs, control verbs, etc. is by invoking worlds where, for instance, water is not H2O, or where Hesperus is not Phosphorus. Presumably these are not metaphysically possible worlds, but rather 'logically' possible worlds. (Sometimes metaphysically possible worlds are called 'broadly' logically possible worlds; by 'logically' possible worlds I mean what are sometimes called 'strictly' logically possible worlds.)

Do these exist too, in exactly the same way as the metaphysically possible ones? If yes, then we run into problems. After all, isn't it only the metaphysically possible worlds which can exist? If not, then what is the distinction between metaphysical possibility and mere logical possibility supposed to mean? In fact, if merely logically possible worlds exist just like the metaphysically possible ones then there is no distinction. But there is, of course, a distinction.

At the very least, aren't the metaphysically possible worlds the only ones which could be actual? But if 'actual' is indexical as Lewis thinks, and the logically possible worlds exist on a par with the metaphysically possible ones, then any of these worlds could be actual.

Personally, I think there's just as good reason to admit the existence of logically impossible worlds as there is to admit the existence of possible worlds (though I don't think there's much reason to admit the existence of either).  If we really needed possible worlds, I think we'd need impossible ones too. But if logically impossible worlds are serviceable too then that makes things even worse for the modal realist. After all, what would it mean to say that a logical contradiction actually holds true in a concrete world just like ours? Clearly there are no such concrete worlds, since whatever concretely exists must at least be possible. But even if one resists the need for impossible worlds, the metaphysically possible worlds are a proper subset of the strictly logically possible ones, and it should be clear that these latter are "serviceable" too.

In sum, if Lewis's argument works for the existence of concrete metaphysically possible worlds, then it works for the existence of metaphysically impossible worlds too. But these can't exist concretely; that's the whole point of making some metaphysically possible and others not. Hence, Lewis's argument does not work. This can be taken as either reason to abandon the 'serviceability' criterion of existence, or as reason for rejecting concrete possible worlds. I'm inclined to reject both.

Wednesday, March 18, 2015

"Whatever is Moved is Moved by Another"

In this post I am going to try to defend Aquinas's First Way, specifically against the attacks brought against it by my friend Alex. Alex has written a fine explanation and critique of Aquinas's first and most famous Way, the argument from motion. The paper can be found here. Unlike many attacks on Aquinas's argument, Alex's reading of Aquinas is sympathetic and charitable, and thus at the same time his criticisms are incisive and well-taken. Anyone who wants to fully understand my post should read Alex's paper first; nevertheless, I will summarize some of his most important results.

To be specific, I'm going to defend Aquinas's premise that whatever is moved is moved by another, which we shall call (MOV). I do not claim that Aquinas ever made the defense I am making. In fact, I think the argument I give is in some respects new. But when all is said and done what I am concerned with is whether Aquinas's premise is defensible and true.

First of all, Alex points out that the term 'motion' in scholastic philosophy really means change. And to say that an object is changing with respect to some feature P is to say that it is going from being potentially P to actually P (more on this terminology here). I will take this for granted in everything I say about change. Now, in summary, Alex's main objection to Aquinas's defense of (MOV) is that either it is (a) valid but palpably unsound or (b) all its premises are true yet it is invalid, i.e. does not prove the premise (cf. his paper for details). However, Alex does think that Aquinas can defend the following more modest premise, which David Oderberg attributes to Aquinas:

(ACT) If something changes from being potentially F to being actually F then there must be some actual being that initiates this change.

The problem is that the more modest and highly defensible premise (ACT) is not equivalent to (MOV), leaving (MOV) undefended and the First Way ultimately uncompelling.

Before I present my argument in favor of Aquinas's (MOV), we need some definitions. First, the definition of an external object, (EXT):

(EXT) x is an object external to y just in case x is not y and x is not a part of y. [def.]

Let's also define what I will call the notion of change per se. (This is my own terminology.) Intuitively, something changes something else per se if it is the most immediate and fundamental efficacious cause of the change, [or the only sufficient cause such that you can't get any 'closer' to the change]. For instance, my hand pushes a stick which pushes a rock; the idea is that the stick, specifically its tip, is what changes the location of the ball per se. Here is a somewhat more formal definition of changing per se, which we will call (CPS):

(CPS) x is a cause per se of a change in something y with respect to feature P by action E just in case (i) x changes y with respect to feature P by action E, (ii) if at the same time as action E there is an action F of some parts of x, and these parts also change y with respect to feature P by action F, then the action F taken alone is not sufficient for changing y with respect to P, and (iii) x's action E taken alone is causally sufficient for changing y with respect to P [def.]

This definition can be made more precise, but the concept should be somewhat clear. The idea behind what I've called change per se is that whatever changes something else per se is the thing that changes y in the most immediate sense and a sense more proper than other things. So, for instance, take the following objects: Me, my arm, my arm's atoms, and a stick. When my arm changes the location of the stick, I can be said to change the location of the stick; however, I cannot be said to change its location per se, since, arguably, if somehow my arm persisted in its motion without the rest of my body (maybe by a miracle it was detached and could float, pushing things around), it would still be sufficient for the stick's changing with respect to its location (contra (ii)). On the other hand, arguably, my arm, or at least some part of it, changes the stick's location per se by its motion, since clearly it can be said to be changing the stick's location, thus satisfying (i). It arguably satisfies (iii) for the reasons stated, and it arguably satisfies (ii) since intuitively if you removed most of the arm but left a chunk of it or a few of its atoms, and they did the same thing as they did when my whole arm's motion occurred, then they would not be able to bring about the stick's change of location.

Now maybe you will disagree with my example and say that given my definition the arm does not change the stick's location per se. But the example is simply to illustrate what I'm trying to get at. If you deny the example is an example of change per se then you should understand what I mean. Also, I would not be surprised if my definition requires chisholming; nevertheless, I think it is on the right track, and helps get my point across. What is most important is just that we have some intuitive understanding of what I mean by something's changing something else per se.

Now we need the following premises. I will translate them into predicate logic, and from my translations it should be clear which formulas correspond to which English phrases.

(1) For all x, if x is changed with respect to P by something y then there is some actual thing z which changes x with respect to P.

Translation 1: ∀x[∃yCxy→∃z(Az∧Cxz)]

(2) For all x and y, if y is actual and x is changed per se with respect to P by y, then y is either an object external to x or y is a proper part of x.

Translation 2: x∀y[(Ay∧Dxy)→(Eyx∨Pyx)]

(3) For all x, if x is changed with respect to P by something actual y, then there is a z which is actual and changing x per se with respect to P.

Translation 3: x[∃y(AyCxy)→∃z(Az∧Dxz)]

(4) For all x and y, if y is changing x per se with respect to P, then y is changing x with respect to P

Translation 4: x∀y(Dxy→Cxy)

(5) For all x and y, if x is external to y, then x is not identical to y.

Translation 5: x∀y(Exy→x≠y)

(6) For all x and y, if x is a proper part of y, then x is not identical to y.

Translation 6: x∀y(Pxy→x≠y)

Let's examine whether these premises are plausible or not. 4, 5 and 6 can easily be shown to follow from the definitions of 'change per se', 'external object', and 'proper part' respectively, so I will not talk about them any more. 1 is basically just a more precise statement of (ACT), so I won't say too much in its defense, but the premise is eminently plausible: Upon a small amount of reflection it is simply obvious that what is merely potential cannot have any power to bring about something actual. Merely potential chemical reactions do not bring about any actual chemical reactions. So the only thing which can bring about something is something which actually exists, and doesn't merely potentially exist. 

The crucial premises then are 2 and 3. 3 is quite plausible on the face of it. For surely if something is changed at all, then there is something which changes it in the most immediate sense i.e. changes it per se. There must be some most immediate explanation or cause of a change right? If there isn't, then the change can never come about. This seems intuitive enough.

(The intuition is this: There seems to be some sort of infinity problem here, though the problem isn't with an infinite regress but rather with what we can call an infinite "progress" of causes. If there is no immediate cause, there has to always be another cause that's "closer" to the effect, but never one that actually "gets" to the effect. If it isn't clear what I mean I can elaborate.)

What about 2? The idea behind 2 is that some things can truly be said to bring about changes in themselves in some sense, but they can't be said to bring about per se changes in themselves; properly speaking, it is the parts which are bringing about the change in the whole. For instance, dogs can move themselves only because their legs do. So, the only thing which can bring about a per se change in something is either something external to it or else one of its parts.

Suppose to the contrary that the cause x of the per se change in y with respect to P is not one of y's parts and is not something external to y. Then since clearly whatever is not a proper part of y and is not external to y is identical to y, it follows  x = y. So y brings about a per se change in y. Now either (a) some of the parts of y bring about the change in y or (b) none do (either way, definitely no parts bring it about per se, as per our assumption).

Assume (a). If none do, then y's parts remain completely the same, yet there is a change in y. But surely y taken alone is not sufficient for explaining the change in y, and thus y does not cause a per se change in itself! After all, how could y change itself with no external influence and no action of any of its parts at all? It would have to be a spontaneous causa sui! So on the supposition that the parts do not act in any way so as to bring about the change in y, it follows y cannot be a per se cause of a change in itself. Since we assumed however that y does cause a per se change in itself, it follows we must reject this supposition and conclude that some of the parts do in fact bring about a change in y. In other words, we must reject (a) and assume (b).

Assume (b). Suppose on the other hand that some of the parts do help cause the change in y. By the definition of per se change, the action of these proper parts of y is not sufficient for explaining the change in y; but nevertheless the action of y taken apart from any external cause is. This seems to make little sense; y still appears to be acting as a causa-sui, since it is still causing a change in itself at least in part independently of the action of its parts. Since this is impossible--nothing can be a self-cause except by the action of its parts--we must conclude that the parts do not help cause the change in y. Thus (b) is false.

Since both (a) and (b) are false, and either (a) or (b) must be true given our assumption that y causes a per se change in itself, we must reject our assumption that y caused a per se change in itself. But if that is the case, then given that there is no external cause of y then x (the cause of the change in y) must be a proper part of y, as we set out to prove.

So much for premises 1 and 2 then. Now, given that all the above premises 1-6 are true we can prove:

(7) For all x, if x is changed with respect to P by some y, then x is changed by some z non-identical to itself.

Translation 7: x[∃yCxy→∃z(Dxz∧x≠z)]

I won't explain the proof here; instead, for anyone who doubts me, I have attached a formal proof below. From 7 and 4 of course it can be shown quite easily that whatever is changed with respect to P is changed with respect to P by some non-identical z: That is to say, whatever is changed is changed by another. Hence, given my 1-6, Aquinas's premise is secure.

Proof of 7:

[Note: If you can't see the proof, right click and either open in a new tab or else save the image and zoom in with some image viewer. I did the proof rather quickly so it is not the most elegant and could be done in fewer steps, but it gets the job done.]